Formal solutions to non-linear ODE

1994 ◽  
pp. 83-101
Author(s):  
Werner Balser
Keyword(s):  
Non Linear ◽  
Linear Ode ◽  
Formal Solutions

Helicoidal surfaces with prescribed curvatures in a conformally flat 3-space

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Chul Woo Lee ◽  
Jae Won Lee ◽  
Dae Won Yoon
Keyword(s):  
Mean Curvature ◽  
Extrinsic Curvature ◽  
Conformal Factor ◽  
Second Order ◽  
Conformally Flat ◽  
Non Linear ◽  
Linear Ode ◽  
Helicoidal Surfaces ◽  
Conformally Flat Metric

Abstract In this paper, we study a conformally flat 3-space 𝔽 3 {\mathbb{F}_{3}} which is an Euclidean 3-space with a conformally flat metric with the conformal factor 1 F 2 {\frac{1}{F^{2}}} , where F ⁢ ( x ) = e - x 1 2 - x 2 2 {F(x)=e^{-x_{1}^{2}-x_{2}^{2}}} for x = ( x 1 , x 2 , x 3 ) ∈ ℝ 3 {x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}} . In particular, we construct all helicoidal surfaces in 𝔽 3 {\mathbb{F}_{3}} by solving the second-order non-linear ODE with extrinsic curvature and mean curvature functions. As a result, we give classification of minimal helicoidal surfaces as well as examples for helicoidal surfaces with some extrinsic curvature and mean curvature functions in 𝔽 3 {\mathbb{F}_{3}} .

scholarly journals Chemically reacting on MHD boundary-layer flow of nanofluids over a non-linear stretching sheet with heat source/sink and thermal radiation

2018 ◽  
Vol 22 (1 Part B) ◽  
pp. 495-506 ◽  
Author(s):  
Oluwole Makinde ◽  
Fazle Mabood ◽  
Mohammed Ibrahim
Keyword(s):  
Boundary Layer ◽  
Heat Source ◽  
Chemical Reaction ◽  
Stretching Sheet ◽  
Skin Friction Coefficient ◽  
Convective Boundary ◽  
Similarity Transformations ◽  
Non Linear ◽  
Linear Ode ◽  
Source Sink

In this paper, steady 2-D MHD free convective boundary-layer flows of an electrically conducting nanofluid over a non-linear stretching sheet taking into account the chemical reaction and heat source/sink are investigated. The governing equations are transformed into a system of non-linear ODE using suitable similarity transformations. Analytical solution for the dimensionless velocity, temperature, concentration, skin friction coefficient, heat and mass transfer rates are obtained by using homotopy analysis method. The obtained results show that the flow field is substantially influenced by the presence of chemical reaction, radiation, and magnetic field.

scholarly journals An application of cubic B-Spline finite element method for the Burgers` equation

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 195-202
Author(s):  
Emine Aksan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Burgers Equation ◽  
B Spline ◽  
Linear Pde ◽  
Non Linear ◽  
Spline Finite Element ◽  
Linear Ode ◽  
Element Method ◽  
Spline Finite Element Method

It is difficult to achieve exact solution of non-linear PDE, directly. Sometimes, it is possible to convert non-linear PDE into equivalent linear PDE by applying a convenient transformation. Hence, Burgers? equation replaces with heat equation by means of the Hope-Cole transformation. In this study, Burgers? equation was converted to a set of non-linear ODE by keeping non-linear structure of Burgers? equation. In this case, solutions for each of the non-linear ODE were obtained by the help of the cubic B-spline finite element method. Model problems were considered to verify the efficiency of this method. Agreement of the solutions was shown with graphics and tables.

scholarly journals Reproducing kernel functions and homogenizing transforms

2021 ◽  
Vol 25 (Spec. issue 1) ◽  
pp. 9-18
Author(s):  
Elif Nuray Yildirim ◽  
Ali Akgul ◽  
Mustafa Inc
Keyword(s):  
Differential Equations ◽  
Reproducing Kernel ◽  
Physical World ◽  
Vital Role ◽  
Higher Order ◽  
Kernel Functions ◽  
Non Linear ◽  
Linear Ode ◽  
Reproducing Kernel Theory ◽  
Higher Order Differential Equations

A lot of problems of the physical world can be modeled by non-linear ODE with their initial and boundary conditions. Especially higher order differential equations play a vital role in this process. The method for solution and its effectiveness are as important as the modelling. In this paper, on the basis of reproducing kernel theory, the reproducing kernel functions have been obtained for solving some non-linear higher order differential equations. Additionally, for each problem the homogenizing transforms have been obtained.

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